Pavement material microwave density measurement methods and apparatuses

ABSTRACT

A method of obtaining a material property of a pavement material from a microwave field generally includes generating a microwave frequency electromagnetic field of a first mode about the pavement material. The frequency response of the pavement material in the electromagnetic field can be measured, such as by a network analyzer. The measurement of the frequency response permits correlating the frequency response to a material property of the pavement material sample, such as the density. A method of correcting for the roughness of a pavement material divides the pavement into a shallow layer and a deep layer. Two planar microwave circuits measure the permittivity of the shallow and deep layer. The permittivities are correlated to correct for roughness. An apparatus for obtaining the density of a pavement sample includes a microwave circuit and a network analyzer. The network analyzer measures the frequency response to determine the density of the pavement material.

CROSS-REFERENCE TO RELATED APPLICATIONS

This application claims the benefit of the filing date of U.S.Provisional Patent Application Ser. No. 60/514,031 filed Oct. 24, 2003,which is hereby incorporated by reference.

FIELD OF THE INVENTION

This invention relates to the measurement of the quality of a pavementmaterial such as soil, sand, aggregate, asphalt, and cement, and moreparticularly to a test instrument and method for measuring orcorrelating the density of a pavement sample using microwave bandwidths.The method is based on precise measurements of the real and imaginaryparts of the permittivity in the frequency or time domain. It also hasapplications for measuring the free water content of constructionmaterials.

BACKGROUND OF THE INVENTION

Pavement materials, such as soil, sand, aggregate, asphalt, and cement,require quality testing for qualities such as moisture and density.Destructive tests and nondestructive tests are used throughout theindustry for determining these qualities. In laboratory destructivetests, cylindrical samples are prepared, typically with a gyratorycompactor, and various material properties are studied to determine thebest mix design for a pavement. In field destructive tests, cylindricalsamples are cored from test strips, newly constructed roads, or existingroads. The material properties of these samples are then used toevaluate whether the test strip or the new pavement meets the designcriteria and whether the existing road is in good operating condition orin need of repairs.

Currently, several methods are used for measuring the density ofcylindrical samples: dimensional analysis, the water displacementmethod, the paraffin coated method, and the para-film-covered method. Ineach case, the bulk density of a sample is derived by, as in thedefinition, dividing the dry sample mass by the estimated sample volume.All methods require a balance with a sensitivity of 0.1 g. to measurethe mass of the sample. In the dimensional analysis method, samplevolume is determined from the radius and thickness (height)measurements. Here, many readings of the radius and thickness of thesample are made using a either manually vernier caliper or automaticallyusing a laser system. The average values of the radius and the thicknessare then used to calculate the sample volume.

Other methods use the Archimedes Principle for determining the samplevolume. These methods require a large container filled with clean water.The water temperature should be monitored and controlled at a specifictemperature, e.g. at 25° C. At one stage of the test, the sample is keptimmersed in water for approximately 4 minutes and the weight of thesample, while suspended in water, is recorded. In the “paraffin-coated”method, after determining the dry weight of the sample, a thin coatingof paraffin is applied to cover the entire surface area of the sample.Then, the sample is weighed again in air. Finally, the sample is weighedwhile immersed in water. More details can be found in standards ASTM D2726 for the water displacement method and ASTM D 1188 for theparaffin-coated method.

Nondestructive field measurements of asphalt are typically accomplishedwith nuclear gauges. Nuclear radiation gauges have been widely used formeasuring the density of soil and asphaltic materials. Such gaugestypically include a source of gamma radiation which directs gammaradiation into the test material, and a radiation detector locatedadjacent to the surface of the test material for detecting radiationscattered back to the surface. From this detector reading, adetermination of the density of the material can be made.

Nuclear gauges, however, require a high degree of training andradiological management for the operators of these gauges. Therefore, itwould be desirable to obtain accurate field measurement gauges withoutthe use of nuclear gauges.

SUMMARY OF THE INVENTION

Methods and apparatuses for obtaining the density of a pavement materialsample by microwave electromagnetic analysis are therefore provided.According to one embodiment of the invention, a method of obtaining amaterial property of a pavement material from a microwave fieldgenerally includes generating a microwave frequency electromagneticfield of a first mode about the pavement material. The frequencyresponse of the pavement material in the electromagnetic field can bemeasured, such as by a network analyzer. One example of measuringfrequency response may include obtaining scattering parameters for thefrequency response although other frequency response parameters may beemployed. The measurement of the frequency response permits correlatingthe frequency response to a material property of the pavement materialsample, such as the density. Generally, the permittivity of a materialpermits direct correlation to the density of a material, andpermittivity is a convenient way to assess density. However, thefrequency response may be used to directly or indirectly calculate thedensity by other methods.

In further more particular embodiments of the invention, a method mayalso include calibration techniques. Calibration may be accomplished bygenerating a microwave frequency electromagnetic field of a first modeabout a calibration material. The calibration material should have knownphysical properties such as density, volume, specific gravity, orporosity. Similar to the measurement method, a frequency response of thecalibration material may be determined. The frequency response of thecalibration material may be correlated to the known physical propertiesof the calibration material, thus providing calibration curve for thefrequency response of the pavement material.

A method of correcting for the roughness of a pavement material is alsoprovided. Generally the roughness may permit dividing the pavement intoa shallow layer, the rough part, and a deep layer. Accordingly, a methodof determining the permittivity of a pavement material having a shallowlayer and a deep layer generally includes measuring a pavement materialwith first and a second planar circuit permittivity sensors. The firstplanar circuit sensor measures the permittivity in a sample volumecorresponding to at least a portion of both the shallow layer and thedeep layer, the second planar circuit sensor measures permittivity in asample volume corresponding at least a portion of the shallow layer. Acalibration data set may be obtained for the first and second planarcircuit sensors by calibrating to a plurality of known pavement materialpermittivities. The first planar circuit sensor measures thepermittivity over the pavement material to permit determining a firstmeasured permittivity. The second planar circuit sensor over thepavement material to permit determining a second measured permittivity.The first permittivity and second permittivity are correlated with thecalibration data to determine the permittivity of the pavement material.

Additionally, embodiments of the invention include an apparatus forobtaining the density of a pavement sample. The apparatus generallyincludes a microwave circuit of a size and shape to generate anelectromagnetic field about a pavement material. A network analyzer isinterconnected to the microwave circuit to generate a microwave input tothe circuit for generating the electromagnetic field. The networkanalyzer should be capable of measuring at least one scatteringparameter. From the measured scattering parameter, the apparatus maydetermine the density of the pavement material.

BRIEF DESCRIPTION OF THE SEVERAL VIEWS OF THE DRAWINGS

FIG. 1 is an apparatus for obtaining the density of a pavement materialaccording to one embodiment of the invention;

FIG. 2 is a diagram of an apparatus for obtaining the density of apavement material according to one embodiment of the invention;

FIG. 3 is a resonant cavity waveguide including a material sampleemployed for obtaining the density of a pavement material according toone embodiment of the invention;

FIG. 4 is a plot of finite element analysis and theoretical solution ofthe cavity and sample of FIG. 3 according to methods of one embodimentof the invention;

FIG. 5 is a ratio, K, of the finite element analysis for correction tothe theoretical solution as a function of dielectric constant accordingto methods of one embodiment of the invention;

FIG. 6 is a plot of frequency sweep of the cavity of FIG. 3 and a sampleillustrating the resonant frequency;

FIG. 7 is a plot of permittivities for a series of samples foundemploying methods according to one embodiment of the invention for aseries of samples having known densities;

FIG. 8 is a plot of densities for a series of samples found according tomethods of one embodiment of the invention compared to densities foundby independent methods;

FIG. 9( a) is an electrical equivalent diagram of a tuned resonantcavity and FIG. 9( b) is an electrical equivalent diagram of a detunedshort microwave cavity according to one embodiment of the presentinvention;

FIG. 10 is a smith chart plot of impedance of a detuned cavity accordingto one embodiment of the present invention;

FIG. 11 is a plot of voltage standing wave ratio of a cavity accordingto one embodiment of the present invention;

FIG. 12 is a smith chart plot of impedance of a cavity according to oneembodiment of the present invention;

FIG. 13 is a plot of frequency sweep of a cavity and a sampleillustrating the resonant frequency found according to methods of oneembodiment of the invention;

FIG. 14 is a plot of densities and resonant frequencies from differentsamples according to methods of one embodiment of the invention;

FIG. 15 is a plot of slotline impedance and density for a planarmicrowave circuit according to one embodiment of the present invention;

FIGS. 16( a) and (b) are microstrip microwave elements for obtaining thedensity of a pave material according to one embodiment of the invention;

FIG. 17 is a microstrip microwave element for obtaining the density of apavement material according to one embodiment of the invention;

FIG. 18 is a plot of resonant frequency response to changes in densityof samples tested by a microstrip microwave element for obtaining thedensity of a pavement material according to methods of one embodiment ofthe invention;

FIGS. 19( a) and (b) are plots of frequency and quality factor responsesfor samples of different densities tested by a microstrip microwaveelement for obtaining the density of a pave material according tomethods of one embodiment of the invention;

FIG. 20 is a microstrip microwave element disposed over a pavementmaterial for obtaining the density of the pavement material according tomethods of one embodiment of the invention;

FIG. 21 is a plot of permittivity measurement of a deep layer of apavement material to obtain a surface roughness correction according toembodiments and methods of the present invention;

FIG. 22 is a plot of ratios of shallow and deep measurements of apavement material to obtain a surface roughness correction according toembodiments and methods of the present invention; and

FIG. 23 is a plot of ratios of shallow measurements and deep measurementof a pavement material sample to obtain a surface roughness correctionaccording to embodiments and methods of the present invention.

DESCRIPTION OF THE INVENTION

The present inventions now will be described more fully hereinafter withreference to the accompanying drawings, in which some, but not allembodiments of the inventions are shown. Indeed, these inventions may beembodied in many different forms and should not be construed as limitedto the embodiments set forth herein; rather, these embodiments areprovided so that this disclosure will satisfy applicable legalrequirements. Like numbers refer to like elements throughout.

Pavement materials, asphalt in particular, are heterogeneous mixtures ofair, rocks, and binder. Each of these materials has a particularpermittivity (i.e. the dielectric constant) associated with its abilityto be polarized by an electric field. The permittivity is linearlyrelated to the polarizability and is a complex quantity. Thepermittivity is generally complex having real, ε′, and imaginarycomponents, ε″, representing energy storage and energy loss respectivelyto a propagating electromagnetic wave. Typically, when speaking of thedielectric constant one is referring to the real part of thepermittivity.

Air has a dielectric constant of 1.0, asphalt binders have dielectricconstant between 2.0 and 4.0, and rocks have dielectric constants thatvary, but granite is about 4.0. In a moisture measuring mode, themicrowave device would be used to measure the volume or mass percent offree water. Free water has a dielectric constant of 80 or less dependingon the temperature. Bound water has a dielectric constant near that ofice of about 3 or 4. For a complicated heterogeneous mixture ofmaterials like soil and water, the permittivity can be a strong functionof frequency when measured swept over a 1 Ghz bandwidth. For HMA (hotmix asphalt), the aggregate is dried, and any residing water is bound tothe aggregate. Bound moisture has a dielectric constant near 3 or 4.This is close to the permittivity of the binder and dry aggregates, andis not problematic as long as the mix remains consistent.

The measured permittivity of a material depends on the frequency and thetype of charge carrier that is being displaced by an applied electricfield. These charge carriers are displaced thus forming a net dipolemoment. The charges can be electronic, atomic, polar and ionic. Inasphalt, all of the above mechanisms contribute to the apparentdielectric constant. However, the main contributions are due to thepolar and ionic responses, on a per unit volume basis. Additionalsolvents or impurities such as water will increase these contributionsand the apparent dielectric constant. For low frequencies, the heavyions respond and the Maxwell-Wagner effect makes the asphalt appearstrongly polar and temperature dependent. At microwave frequencies, thiseffect is nonexistent. The Maxwell-Wagner effect also allows for thequantification of soil moisture and density of the heterogeneousmaterials.

The behavior of polar molecules in the asphalt follows afrequency-temperature response that can be modeled with a modified Debyeequation. The dispersion in the microwave region is significantlydecreased as a result of the reduction of the relaxation frequency. Thisreduction is anticipated due to the heavy asphalt molecules, and becauseof the asphalt bonding to the aggregate, which is of course the purposeof adding the binder in the first place.

Although the asphalt liquid is characterized as non-magnetic, theaggregates that make up the pavement mix could very well have a magneticmoment leading to a complex permeability μ′−jμ″. Here, μ′ is the realpart representing the ability to store magnetic energy, and μ″ is theimaginary part representing loss. Since both ε and μ can slow anelectromagnetic wave, and both increase on a per unit volume basis withcompaction, the permeability could play a role in the measurement of theasphalt density, but only if a substantial magnetic field H exists inthe material. Likewise, ε″ and μ″ will increase the energy loss withcompaction. It has been suggested to add lossy ferrite oxides to anasphalt mix in order to increase the efficiency of microwave heating.The asphalt can be heated during application to maintain temperature, orduring repairs or removal.

Referring to FIG. 1, one method of measuring real and imaginary parts ofpermittivity is defined by ASTM Standard D2520-95, which is herebyincorporated by reference. The test method of D2520 is based onmicrowave measurement of the complex scattering (S) parametersthroughout a specimen in a resonant transmission line or closed cavity.In this technique, the sample is placed inside a transmission line orcavity and is the device under test (DUT) 10. The methods hereinincorporate these teachings and further can be extended to leaky typeresonators. The measurements may be implemented by a network analyzer 20analyzing the wave propagation through the material specimen usingS-parameter analysis. They can be implemented also by analyzing the wavepropagation through a transmission line or waveguide and interactingwith the material at the boundary of the sensor.

S-parameter analysis techniques may be accomplished by a variety ofcommercially available broadband network analyzers, such as the HP(Agilent) 8753 Network Analyzer manufactured by Agilent Technologies,Inc. of Palo Alto, Calif., as shown in FIG. 1. Network analyzers 20 arewell known to those of ordinary skill in the art for measuring thevoltage standing wave ratio, complex reflection/transmissioncoefficients, impedance and admittance matrix parameters, and return andinsertion loss of a device under test. The setup typically includes thenetwork analyzer 20, a transmission/reflection test set 22, and thedevice under test 10, which is described in various embodiments below.These instruments are capable of both frequency and time domainanalysis. The conversion between these two domains is accomplished usingthe Fast Fourier Transform commonly known as the FFT. For example, animpulse or step response can be analyzed for dispersion and groupvelocity. The velocity is inversely related to the real part of thedielectric constant and in fact proportional to 1/√ε′(ω) whereas theloss is related to the imaginary part ε″(ω). Hence, both the real andimaginary parts can be frequency dependent, as one skilled in the artwould know. These days it is now possible to economically build anddesign one port and full two port network analyzers that are compact andefficient on battery power.

FIG. 2 is a simplified diagram of a network analyzer 20, including afrequency generator 8, illustrating the switching network 12, 14 anddirectional couplers 16, 18 for a full two port system to a DUT 10.There are many designs that could be used for the network analysisincluding scaler and vector network analyzers, some use simple diodedetectors as can be found in publications like “Microwave ImpedanceMeasurement by P. I. Somlo, BS J. D. Hunter, Peter Peregrinus Lt.,London UK. 1985, or Microwave Measurement Edited by A. e. Bailey, PeterPeregrinus Lt., London UK. 1985. Reflectometers, VSWR bridges, andnetwork analyzers can be based on bridge circuitry for lowerfrequencies, and couplers for the higher frequencies.

Impedance, permittivity, permeability, complex permittivity, and complexpermeability of a material can be measured using S-parameter analysis,such as described in Agilent AN 154 S-parameter Design Application Note,which is hereby incorporated by reference. As S-parameters relate ratiosof RF network inputs to outputs, they can be used to measure the ratiosof RF network inputs and outputs for a material placed within a closedchamber or near an open sensor system. An incident RF electromagneticwave is actively excited and the system responds according to theboundary conditions and constitutive relations of the material. Forexample, S₁₁ is a reflection measurement from the device under test thatmay yield return loss. S₁₂ is a transmission measurement through thedevice under test that may yield insertion loss. As such, S₁₁ and S₁₂may be measured as a ratio of the RF input and output, in magnitude andphase. Hence these embodiments of the invention employ a measuringsystem comprising a wideband microwave sensor for exciting the DUT, acoupler, and a measuring circuit allowing scattering (S) parameters,impedance (Z) matrix parameters, or admittance matrix (Y) parameters anda data processing unit with display and software. From these,calculations of stored energy in a near field, parameters such aspropagation constants α, β, resonant frequency and Q, and phase andgroup velocity can be deduced. From these (not limited to thesespecifically) a resulting property of the sample is found. The systemuses a microwave sweeping oscillator to permit measuring the parameterssuch as the scattering parameters and other matrix parameters as afunction of frequency, such as to locate a resonant frequency, Q,maximum or minimum of S₁₁ or S₂₁ as a function of frequency. The displayand software could be integrated into a single unit much like a Troxler3440 Nuclear gauge. This instrument incorporates downhole as well assurface measurements. The processing unit and display could be linked toa PDA using a wire or wireless connection. The software would includeall the basic commands in addition to project management programs.

While material measurements have been described in conjunction withnetwork analyzers to determine parameters such as impedance,permittivity, permeability, complex permittivity, and complexpermeability by using S-parameters, it will be understood by those ofordinary skill in the art that other measurement systems utilizing othermaterial analysis techniques may be used in conjunction with thesemethods and the device under test described herein. In fact, anyelectromagnetic wave analysis procedures that permit determinations ofpermittivity and complex permittivity either via software or manualmethods, may be implemented in conjunction with the present inventionwithout departing form the spirit or scope of the claims.

Resonant Cavity Sample Gauge

In the asphalt pavement construction industry, the cylindrical asphaltcore is a common sample geometry. Accordingly, asphalt cylinders mayadvantageously be used in a resonant cavity to determine permittivityproperties. According to one embodiment of a method of determining thedensity of an asphalt sample, the permittivity analysis may yielddensity measurements. These results could be used to calibrate fieldinstruments. It should be noted that while a rectangular resonant cavityis described below as an appropriate measurement device for an asphaltsample, other resonant devices are known to those of ordinary skill inthe art and may be substituted accordingly. Other waveguide resonatorsinclude cylindrical, reentrant, Fabry Perot, or the devices made frommicrostrip, slotline, coplanar waveguide, coplanar strips, surfacewaveguide, dielectric resonator and any of a number of resonatingstructures and modes such as TEM, quasi-TEM, TE, TM, or hybrid HE modes.The measurements could be made in the reflection mode, transmission modeor combinations thereof. The structures could have loads to satisfycertain boundary conditions such as open, short, match, or even aresonant load such as a short or resonant dipole. Here, the dipole wouldreact with the near field and the impedance would shift the resonantfrequency of the feed.

These could be realized using a microstrip patch antenna, or even anantenna array. In any event, the microwave radiation will interact inthe near field with the material under test for contact devices, and inthe far field for non-contact devices or deeper measurements.

As described herein, it should also be noted that the sample is notnecessarily small compared to a wavelength, but asphalt samples have adielectric constant typically less than about 8 and are constant inshape. As such, a constant shape of both calibration samples andmeasured samples permits shape error to be corrected. The large size ofthe sample, as compared to the wavelength, can be corrected for with acorrection factor K. The correction factor may be found using a finiteelement analysis or any number of numerical computational techniques,discussed in more detail below. If it is a common shape such as a smallcube or rod it is possible to obtain corrections using rigorousmathematics with appropriate boundary conditions, but in general, theFinite Element or other numerical techniques are useful for anygeometry. Other numerical techniques include finite difference, finitedifference time domain, method of moments, etc. Therefore, the abilityto disregard shape, the ability to correct for size, and the relativelylow dielectric constant permits perturbation techniques or in general“wave like propagation” methods to be used to measure the permittivityof asphalt samples or surfaces. The sensors can be used for spotchecking an asphalt or soil surface, or a surface wave could be launchedand detected covering larger areas. It is well known that a surface wavehas a penetration depth inversely proportional to the frequency ofexcitation. This device could be a slow wave structure such as acorrugated slab in contact with the medium of interest. The propagationconstant as a function of frequency could then yield information of thequality vs. depth.

The following illustrates one method according to one embodiment of theinvention for obtaining the density of an asphalt sample from apermittivity analysis. Referring now to FIG. 3, a rectangular TE₁₀₁ moderesonant cavity 34, which is may be one embodiment of the DUT in FIG. 1,was used to obtain measurements of an asphalt sample 32. Thisrectangular resonant cavity has dimensions of 0.9×0.4×0.19 meters. ASTMstandard D-2520 is a standard for small samples that meet certainboundary conditions that may be employed with boundary valuemodifications. In the standard, the sample is presumed to be thin andextends all the way through the cavity. Such a sample allows for easysolutions of Maxwell's Equations and the necessary energy analysis.First, the small sample does not “perturb the fields” hence you canassume that the field configuration is exactly the same before and afterinsertion. Furthermore you can calculate the fields of the empty cavity,and use the same solutions for the loaded cavity. Also, with this setupand mode, the sample is always tangent to the fields and there are noperpendicular fields interacting with the sample. This assures that thefield in the sample is exactly the same as with the empty cavity.

In FIG. 3, however, there is a large sample 32 that sits in the bottomof the cavity 34. From boundary conditions the field inside the samplewill be different from the empty cavity, as we have normal components.Also since the sample 32 is about 0.15 m wide and 0.1 m tall it is nolonger electrically small compared to a wavelength. So the ASTM boundaryconditions do not apply if the sample does not extend fully from top tobottom, or was not always tangent to the electric field. It would beconvenient to have air gaps between the cavity 34 and the sample 32 sothat it inserted easily, and this would require some correction as well.

For an empty, air filled, rectangular cavity the resonant frequency isfound by solving Maxwell's Equations with the appropriate boundaryconditions, so every resonant structure will have its own uniquerelationship for resonant frequency.)

The fundamental mode for the rectangular cavity is

$\begin{matrix}{f_{101} = {\frac{c}{2\pi\sqrt{\mu_{o}ɛ_{o}}}\sqrt{\left( \frac{\pi}{\text{width}} \right)^{2} + \left( \frac{\pi}{\text{length}} \right)^{2}}}} & (1)\end{matrix}$For this example cavity 34, the empty resonant frequency isapproximately 428 MHz. It is known from perturbation theory that

$\begin{matrix}{\frac{\delta\; f}{f} = {{\frac{f_{2} - f_{1}}{f_{2}} + {\frac{j}{2}\left( {\frac{1}{Q_{2}} - \frac{1}{Q_{1}}} \right)}}\mspace{34mu} = {{- \frac{\left( {ɛ_{r} - 1} \right)}{2}}\frac{\int_{V_{Sample}}{{{E_{1} \cdot E_{2}}}^{2}{\mathbb{d}v}}}{\int_{V_{Cavity}}{{E_{1}}^{2}{\mathbb{d}v}}}}}} & (2)\end{matrix}$(See R. F. Harrington, “Time Harmonic Electromagnetic Fields”, McGrawHill Book Co., 1961.) Where E₁, f₁, and Q₁ are the electric field,frequency and quality factor for the empty cavity and E₂, f₂ and Q₂ arefor the cavity containing the sample 34, while V_(cavity) and V_(sample)are the volumes of the cavity 34 and sample respectively. The qualityfactor Q is the ratio of energy stored in the system to that which isdissipated per cycle. For a transmission type resonator, it is measuredusing the 3 db bandwidth and the resonant frequency of theconfiguration. The quality factor is defined for any resonant systemwhether it is a waveguide cavity, a microstrip resonator, a resonanttransmission line or a cavity backed antenna.

For an unperturbed TE₁₀₁ mode cavity the electric field intensity, E, is

$\begin{matrix}{E_{y} = {E_{o}\sin\frac{\pi\; x}{a}\sin\frac{\pi\; z}{d}}} & (3)\end{matrix}$where a is the width of the cavity and d the length in meters.Assuming for small samples that E₁=E₂, Equation 3 may be substitutedinto Equation 2 resulting in the solution for real and imaginary partsof permittivity, as:

$\begin{matrix}{{\frac{f_{2} - f_{1}}{f_{2}} + {\frac{j}{2}\left( {\frac{1}{Q_{2}} - \frac{1}{Q_{1}}} \right)}} = {{- \frac{ɛ^{\prime} - 1 - ɛ^{''}}{2}}\left( \frac{4V_{sample}}{V_{cavity}} \right)}} & (4)\end{matrix}$Such that

$\begin{matrix}{ɛ^{\prime} = {{\frac{f_{1} - f_{2}}{2f_{2}K}\frac{V_{cavity}}{V_{sample}}} + 1}} & (5) \\{ɛ^{''} = {{\frac{1}{4}\frac{V_{cavity}}{V_{sample}K^{2}}\left( {\frac{1}{Q_{2}} - \frac{1}{Q_{1}}} \right)} = {\sigma\text{/}\omega\; ɛ_{o}}}} & (6)\end{matrix}$where σ represents the conductivity of the asphalt sample and K is thegeometrical correction factor and accounts for the change in theelectromagnetic fields with shape, permittivity and orientation. For asmall sample whereby the tangential E field is continuous across theboundary and throughout the sample K=1. However, for larger samples orsamples not oriented parallel with the fields, K must be determined byclosed form or using a numerical method such as finite element analysis.In general, the electric field E₂ is only found in context with theknowledge of the sample geometry, material properties and fieldorientation. The sensitivity is maximized when the E field is parallelto the sample boundary, but the analysis works for normal fields orcombinations thereof.

The finite element analysis requires modeling a “virtual” sample havingknown dielectric properties and dimensions. The analysis thereforeyields the complex frequency response of the system. For example, thecomplex propagation constant, resonant frequency and loss effects can beobtained. The finite element results may then be compared to actualmeasurements, and the parameters in the finite element model adjusteduntil the model produces exactly the same results as the measurementsover a bandwidth. Hence, a solution to the material parameters isindirectly found. This is computationally extensive, and could requirehours to find a solution.

In another approach, curves of resonant frequency or loss vs.permittivity and sample dimension could be found using the computationaltechniques and stored in a computer. Then in the portable device, thesample dimensions could be entered into the computer along with the lossand frequency results, and the complex permittivity could be found.Still, if the sample geometries are of standard shape as in the asphaltindustry, correction factors to closed form equations can be obtained asis the case of Equations 5 and 6. The correction factors could also beobtained experimentally if one possesses standard materials of knownpermittivity and constant geometry from sample to sample to use forcalibration. It was found that the correction factor for larger samplescould be used for smaller samples of the same general shape.

Typically, when a sample dimension is less than about λ/10 it can beconsidered electrically small. In these experiments, the pucks areusually about 150 mm in diameter, and 115 mm high. In the rectangularwave-guide used in these experiments, the first mode resonated at about426 Mhz in air with a wavelength of about 750 mm. So the sample is notconsidered small, and furthermore, it does not extend to the top of thewave-guide. This gap between the sample and the wave-guide roof is anormal E field boundary condition, which will substantially change thefields in this region for ε_(r)>1. This will further remove the solutionfrom the ideal case.

The difference between the ideal solution and the FEM solution with theshort sample puck of material is illustrated in FIG. 4. A finite elementsimulation was carried out using a rectangular cavity of the dimensions0.9144×0.1905×0.3794 m. Inside the cavity and placed at the point ofmaximum electric field (the center for the fundamental mode) was avirtual cylindrical puck with a radius of 75.057 mm and a height of 115mm. In the numerical analysis, the permittivity was changed from 1 to 8.For an air filled cavity, it was found that the numerical analysis gavea resonant frequency f₁=427.6 MHz. With some adjustment of the magneticloop coupling mechanism this agreed exactly with the actual measuredvalue. The program was then run with the different permittivities, andeach time the Eigen-frequencies were recorded. The frequencies are shownin columns 4 and 5 of Table 1 below.

TABLE 1 δf/f₁ (theory δf/f₁ (Finite f₁ (Finite f₁ (theory) ε_(r)′ d <<λ, K = 1) elements) Element) K = 1 1.0 0 0  427.6 Mhz  427.6 Mhz 1.1.00616 .00538 425.31 424.98 1.2 .0123 .0107 423.08 422.40 1.3 .0185.0159 420.89 419.83 1.4 .0246 .0211 418.75 417.33 1.5 .0308 .0263 416.65414.82 2 .06158 .0508 406.94 402.8 3 .1232 .0944 390.70 380.70 4 .1848.1315 377.92 360.90 5 .2463 .1628 637.72 343.10 6 .3079 .1896 359.44326.94 7 .3695 .2127 352.60 312.23 8 .4311 .2327 346.89 298.79In Table 1 δf=f₁−f₂ and the “theory” columns refer to the use ofEquations 5 and 6 with K=1. The correction factor K can be found foreach sample by simply dividing column 3 by column 2. FIG. 5 is a plot ofK vs. ε_(r) for this geometry. In equation form, the correction factorwas found to beK=0.0477+0.9*exp(−0.076ε_(r))  (6 b)As an example using the real part of the permittivity, a calibrationsample was inserted into the cavity. In one example, the calibrationsample was a polyethylene puck of radius 74.95 m, height of 114.5 mm andpermittivity of 2.6 The frequency was swept over the proper band for thefirst resonance and is shown in FIG. 6. Here we see that the resonantfrequency is 400.08 MHz. Substituting this into Equation 5 with K=1yields a dielectric constant ofε_(r)=1+(66.09/2.021)(427.6−400.08)/(2*400.08)or ε_(r)=2.09. Multiplying this by K found from (6b) results in a valueof 2.57. Notice that the height of the polyethylene cylinder was notexactly the same as that which was used in the FE simulation. As long asthe measurement geometry is relatively close to the simulation, goodresults can be obtained. For larger differences in height h, acorrection for the permittivity can be estimated usingε_(r)=(ε_(r)−1)h/115+1.  (6c)This is derived using the fact that the frequency shift is proportionalto the ratio of volumes of the sample and the cavity Vs/Vc.

As a test incorporating the imaginary part of the permittivity it isknown that DELRIN, a well-known plastic, has a complex permittivity of3.1+j0.148. First a small sample was simulated using the finite elementprogram. This sample extended from the floor to the ceiling of thecavity and was only 25.4 mm in diameter. This gave a sample volume of96.528e−3 m³. Since the electric field vectors are always parallel tothe sample, and the sample is small compared to a wavelength, K=1.Equation (5) and (6) gives a resonant frequency of 424.99 Mhz and a Q of1104. As a check of the numerical program, this material was coded upand it was found that the Finite element analysis yields f=424.94 Mhzand a Q of 1056. A near perfect agreement. Next in the model, the Delrinwas expanded in size to have a radius of 75.057 mm and shortened to aheight of 115.0 mm leaving a substantial air gap between the sample andthe cavity ceiling. According to Table 1, this would result in a K of0.7666. Incorporating K into (5) and (6), and backsolving for frequencywe find f₂=389.21 Mhz. Likewise, Equation (6) results in a quality of93.1, both agreeing well with the numerical analysis. Therefore, it wasconcluded that even with large samples, the geometrical correctionfactor derived only from the real part of the permittivity could also beused for the imaginary calculations, and in fact, could be absorbed intothe shape factor.

To summarize, for the above described cavity, FIG. 4 illustrates theTE₁₀₁ response for a finite element analysis in comparison to thetypical ASTM 2520 method where the field E is not perturbed by theinsertion of the sample. FIG. 5, illustrates the correction factor K asa function of dielectric constant. Furthermore, during the course of theanalysis it was observed that the correction factors for both loss andfrequency shift (thus real and imaginary permittivity) were very close.Therefore it deemed acceptable to assume the same correction factor forboth the real part and the imaginary part of the permittivity. As such,the absolute permittivity of the sample may be determined from the aboveequations; even for electrically large and odd shaped samples.

For circuits, the quality measurements require correction for impedanceloading and loss other than from the sample properties and measurements.For an unloaded device,

$\begin{matrix}{\frac{1}{Q_{L1}} = {\frac{1}{Q_{0}} + \frac{1}{Q_{E}}}} & (7)\end{matrix}$where Q_(L1) represents the total sum of all losses, Q_(o) is theinternal energy loss, and Q_(E) is the external energy loss through thecoaxial cable and network analyzer. For a device loaded with a sample,

$\begin{matrix}{\frac{1}{Q_{L2}} = {\frac{1}{Q_{0}} + \frac{1}{Q_{E}} + \frac{1}{Q_{s}}}} & (8)\end{matrix}$Accordingly, the sample Q may be determined by

$\begin{matrix}{{\frac{1}{Q_{L1}} - \frac{1}{Q_{L2}}} = \frac{1}{Q_{S}}} & (9)\end{matrix}$For all three loss mechanisms, the loss equation is

$\begin{matrix}{Q_{L} = \frac{\omega\; L}{R_{s} + R_{sample} + Z_{o}}} & (10)\end{matrix}$where R_(s) represents internal losses for example the surfaceresistance of the resonator, R_(sample) represents the loss of thesample, and Z_(o) represents the impedance of the coaxial cable orpossibly other external losses such as radiation. At resonance, acoupling factor, β, relates Q_(o) and Q_(L) [R. E. Collin “Foundationsfor Microwave Engineering”, McGraw Hill, 1966.]Q _(o) =Q _(L)(1+β)  (11)wherein β is either overcoupled,β=R′/Z _(o)  (12)undercoupledβ=Z _(o) /R′  (13)or critically coupled.β=1  (14)It is therefore advantageous to undercouple such that β<<1, toapproximateQ_(L)≅Q_(s)  (16)

To summarize, the purpose of the cavity measurements of a calibrationsample is to obtain the complex permittivity of mix materials forcalibration of field devices, and to obtain these properties forlaboratory records of the engineering properties of cores. Theengineering properties are related to the quality of the cores throughthe chemistry of the binder (oxidation), water and void content of thecores.

Now when the permittivity of the asphalt sample has been established,the cavity can be calibrated with multiple asphalt samples of a knowndensity. For example, at least two samples of different density but ofthe same mix (permittivity characteristics) are placed in the resonatorto obtain measurements of Δf or ΔQ resulting in a linear equation fordensity. However, since most mixes have similar slopes, a genericcalibration curve could be obtained by finding only an offset of thecalibration curve. Here, only one asphalt sample of particularpermittivity and known density would be needed.

The simple model incorporated the dimensions of the puck, and a variablepermittivity. The different mixes have different “base” permittivitiesthat are reduced by the increase in air voids, as the density decreases.

Density Relations:

As the material is compacted, the dipole density increases and thus bothε′ and ε″ will increase. These increases will decrease the phasevelocity of the electromagnetic wave, and increase the insertion andreturn loss of the system. For a resonant structure, both the resonantfrequency and the Q will decrease with increasing compaction. Offresonance, the magnitude and phase of the reflection and transmissioncoefficients will change in accordance with the characteristics andlength of the waveguiding structure, in this example a resonant cavity.As a result, it is possible to measure density increases using ε′ and/orε″ increases, or decreases in f and/or Q. With these types ofmeasurements, a calibration using calibration samples of known densitiesand consistent shape is necessary. If the calibration is performed usingSUPERPAVE type asphalt cores of radius about 75 mm and heights near 115mm vs. f and Q, then the calibration curves will be restricted to theseshapes and sizes; except for some height variation that could becorrected using (6c). In other words, the smaller Marshall pills couldnot be used without a separate calibration, as their radius is about 50mm. On the other hand, if the density calibrations are defined using thecomplex permittivity, then no matter what the shape of the sample, thecalibration will hold once the permittivity of the sample is measured.

Shape independent calibrations can be incorporated by taking the ratioof Equations (5) and the square root of (6) such as described in U.S.Pat. No. 5,554,935. Incorporating the fact that V_(s)=mass/ρ where ρ isthe bulk density we findρ=(G/(εf√Q)+C)²  (6d)Where C is a constant. With this approach, frequency and Q can be usedwith the same calibration equation for different shapes.Frequency Only Calibration:

37 cores were produced in a Gyratory compactor. These were of thespecies of limestone and granite. There were 6 series of cores. Witheach series, three different void ratios were targeted in the mix byvarying the mass of the materials for a particular compaction height.The height was near 115 mm just like the simulations, and there were twoduplicates for each density.

In the cavity, two measurements were obtained for each corecharacterized by flipping the sample upside down. So each series had 6measurements. In the cavity, the sample is inserted and the frequencyshifts down from the empty 426.7 Mhz value. Analysis at multiplefrequencies is desirable and could be obtained by selecting higher ordermodes or designing a different cavity.

From the finite element analysis, an equation was derived that directlyrelates the resonant frequency to the permittivity andε′=−0.7690+7355.34exp(−0.0194f)where f is in Mhz. Since the frequency shift is directly related to theratio of volumes Vs/Vc, then for different sample heights a correctionfor the permittivity can be estimated using Equation (6c). Theserelationships were used to find the real part of the dielectricconstants of the cores made in the laboratory. The results were thenused to obtain equations relating the permittivity to the density. Thedensities of the cores were obtained using nuclear instruments andmeasuring methods.

As an example, consider the series 2-xxx cores. For the core 2-102, thefrequency easily converts to a dielectric constant. The measurementresults are shown in Table 2 below where height correction wasincorporated using 6c. The X and O represent measurements whereby thecore was physically flipped in the cavity.

TABLE 2 Freq S21 DeltaF QL X 368.5625 −19.4509 0.72594 507.701 O368.3375 −19.8601 0.76284 482.8496 AVG PCF PCF (X) SampleID Height mm(gmb) (nuclear) mass (g) Microwave 2-102 113.900 2.032 126.472 4102.9126.291 Freq S21 DeltaF QL 368.5625 −19.4509 0.72594 507.701 epsX eppsXfact epsO eppsO fact 5.00 0.039064 0.639827 5.028544 0.041223 0.638667For each different mix, a regression analysis was performed to obtainthe calibration curve for that mix vs. the permittivity. Table 3 showsthe results for series 2.

TABLE 3 CORE er′ er″ PCF slope intercept 2-102 5.042 0.039 126.47217.444 38.340 Flip core 5.067 0.041 126.472 0.367 2.088841 2-103 5.0580.035 126.196 0.994 0.570606 5.067 0.039 126.196 2264.960 14.000 2-2015.726 0.042 138.511 737.450 4.558272 5.740 0.047 138.511 2-202 5.7250.040 138.911 5.728 0.042 138.911 2-203 5.722 0.043 138.929 5.739 0.040138.929 2-301 6.060 0.038 144.228 6.043 0.036 144.228 2-303 6.042 0.040143.019 6.062 0.042 143.019 2-305 6.074 0.042 143.890 6.074 0.038143.890The residual is 0.994. A very good value representing a linearrelationship between the microwave dielectric constant and the densityof the cores.

Note that δf or δQ can be used as a function of density, or ε′ and/or ε″as a function of density. Alternatively, ratios of these parameters maybe used such as ε′/ε″. An example is illustrated in FIG. 7 depicting ε′vs. ρ from the scattering coefficient (S₁₂) results of several differentasphalt mixes. Note that FIG. 12 illustrates only the real part ofpermittivity. Also, each mix has different intercepts but similarslopes. Accordingly, it is acceptable to assume for these asphalt mixesthat all of the curves may be approximated to a single slope and thenonly the intercept needs to be found. This approximation may assist toreduce field calibration steps, however, the preferred method is tooffset both the slope and intercept. FIG. 8 illustrates the accuracywhen each mix is calibrated for both slope and intercept.

For a single mix design, both S₁₁ and S₁₂ can be used to correlatebetween frequency shift and density, thus disregarding actualpermittivity values in this case. In the field, measurements of the fand Q values or S₁₁ and S₁₂ can be easily correlated to the density of atest strip using cores or nuclear gauges without direct knowledge fromthe permittivity.

Reflection S₁₁:

When a transmission line is terminated with a complex impedance,reflection measurements are appropriate. As is well known in themicrowave analysis, the impedance is measured at a port, and the phaseof this port (partly determined be the length from the device or load)defines the impedance in a predictable and cyclic manner. In particular,two common equivalent circuits can be obtained and are defined by the“detuned” resonance. One is the detuned short position, and the other isthe detuned open position. Of course the equivalent circuit can take onan infinite number of impedances in between these extremes, but withproper referencing, the characteristics of the device can be extracted.Since the detuned short and open are mathematically dual to one another,the results of one can easily be carried to the other.

A tuned circuit is represented by FIG. 9( a) where a transmission lineimpedance Zo, a coupling resistance Rs and the cavity loss G, andelectric effect is C, and magnetic field effect is inductance L. Theresonator as a detuned short circuit is shown in the FIG. 9( b), and thereal and imaginary parts are proportional to the square of the couplingturns ratio n. Note that far off resonance, the circuit has theimpedance of a short in the ideal case. This feature could be used tocalibrate the systematic error of the network analyzer. For a microwavedevice, other resonances also effect the results. These models representthe response only near a particular harmonic. The impedance of the TE011detuned cavity is shown in the following figure in a Smith Chart format,FIG. 10. Here the reference plane has been shifted to represent a shortoff resonance.

It should be noted that impedance plot 40 of FIG. 10 is nearly circular.Near the resonance, the resistance or loss is constant with frequency,and this loss is modeled as a constant resistance circle on the SmithChart. At resonance, the impedance is real, and is closest to the centerof the chart leading to a minimum VSWR. For a well matched device, thereflection coefficient in dB will have a very sharp dip making it easyto find the resonance frequency.

A plot of VSWR near resonance is shown in FIG. 11. The procedure is tosweep the network analyzer over the pass band of the device and plot theVSWR as shown in FIG. 11.

For reflection measurements, the resonance corresponds to a minimumreflection coefficient or VSWR. As known in the art, it is also possibleto obtain Q measurements using primarily reflection techniques. In thiscase, Δf is found by the difference in frequency between the half powerVSWR frequencies, such that

$\begin{matrix}{{VSWR}_{\frac{1}{2}} = \frac{1 + S_{o} + S_{o}^{2} + {\left( {1 + S_{o}} \right)\sqrt{1 + S_{o}^{2}}}}{S_{o}}} & (17)\end{matrix}$Where S_(o) is the VSWR at resonance with a sample in the cavity, andthe 3 db power points correspond to f₁ and f₂ of the two VSWR_(1/2)points.

When the system is undercoupledβ=1/S _(o)  (18)which is substituted to Equation 11, Q_(o)=Q_(L) (1+β), since

$\begin{matrix}{Q_{L} \equiv \frac{f}{\Delta\; f}} & (19)\end{matrix}$where Δf is defined by the VSWR_(1/2) above; Q_(o) due to intrernallosses only can be found.

As will be recognized to one of ordinary skill in the art, then it isnot necessarily the values of permittivity of the asphalt sample thatare required to be calculated. Rather the density relationship may becompared to any number of parameters measured from the microwavetransmission and reflection loss, frequency shifts, phase shifts, Sparameters, admittance and impedance matrix parameters, etc. That is tosay that the above simply describes one advantageous method to determinethe density of the sample using the reflection and transmissionmeasurements, while the measurements may be related in a number of othermathematically correct methods to the exclusion of having to solve forpermittivity.

The empty cavity in the reflection mode yielded an impedance plot 50 asa function of frequency shown in FIG. 12. Since β<1, the circle ofconstant resistance never crosses the center of the Smith Chart, andβ=1/VSWR=22.2/50=1/2.25. An easy way to find the resonance is to plotthe return loss or S11 in log magnitude and look for the minimum. Thisplot is shown in FIG. 13 and indicates that the frequency of resonanceis 426.7 Mhz. The half power VSWR (Vc) is found to be 7.25, and thiscorresponds to a frequency bandwidth of about 285 Khz for a loaded Q ofQL=1497, and the unloaded Qo=1.444*QL=2162.

Samples were measured using the reflection method. A table of theresults is shown below.

TABLE 4 Frequency Sample Mhz VSWR Vc Bandwidth QL Qo 2-102 368.23 27.7957.63   849 Khz 433 2-202 362.3 64.94 131.9 874.5 414.35 2-301 359.9562.67 127.36 790.5 455.34Since these VSWR's were so high, the cavity was very much undercoupled,and the unloaded Q will be very close to the loaded Q's.

The resonant frequency (i.e., PCF, as shown below) can be plottedagainst the density of the cores for relative measurements as shown inFIG. 14.PCF=914.24+f*−2.14, r ²=0.99988Also, the dielectric constant can be calculated using the resonantfrequency and the finite element model of the cavity and these resultswould be identically equal to the results obtained using thetransmission cavity.

It is also notable that for microwave frequencies the calibration isless susceptible to change with the conductance due to moisture of aparticular type of dielectric or pavement. Hence, calibrations may bemore universal from mix to mix. In the microwave region, instead ofmeasuring a lumped impedance, the reflection or transmission propertiesof the device are obtained. As for frequency dependence, the dispersionwill be minimized as f increases as explained below.

For free polar molecules, the orientation and induced polarizations area strong function of temperature; especially at relatively lowfrequencies. For moist (water containing) mixtures,

$ɛ_{r} = {ɛ_{r\;\infty} + \frac{\left( {ɛ_{r\; s} - ɛ_{r\;\infty}} \right)}{\left( {1 + {j\;{\omega\tau}}} \right)^{\alpha}}}$where α is a correction for a distribution of relaxation times, τ is therelaxation time directly related to the mobility of the molecules,ε_(r∞) is the high frequency permittivity, and ε_(rs) is the lowfrequency or static permittivity. At frequencies well above 1/τ,ε_(r)=ε_(r∞). ε_(rs) decreases with rising temperature due to theincreasing disorder in the molecules, and τ decreases with increasingtemperature. By investigation of the dispersion of the real andimaginary parts of the permittivity from low frequencies to themicrowave band, the moisture effects may be separated from otherattributes such as porosity. For a heterogeneous mixture, thesefrequency effects will be exacerbated above any simple Debye effects ascan be explained by the Maxwell-Wagner theory.

Since asphalt is a “binding” agent and is adsorbed into the aggregates,and since it is a large heavy molecule, the relaxation time is longresulting in relatively low dielectric constants and a low relaxationfrequency. For this reason, microwave analysis of HMA (Hot Mix Asphalt)is less susceptible to temperature variations when compared to the lowerbands. Hence a wideband measurement swept through the microwave band cantherefore result in the separation of moisture and density effects.

U.S. Pat. No. 6,414,497 and U.S. Published Patent Applications20020175691 discuss the temperature effects of asphalt densitymeasurements. In the microwave region, the temperature effects of polarmolecules are negligible. However there will be temperature effects dueto the simple volume expansion of the material under test. This isbecause the method of measurement is based on the dipole moment per unitvolume of material. In this respect, slight temperature corrections tothe microwave results could be useful for the most accurate results inquality assurance of paving materials. These linear corrections could berelated to the base temperature of the asphalt. For instance, thisexpansion is confirmed by in ASTM standard D 4311-96, “Standard Practicefor Determining Asphalt Volume Correction to a Base Temperature”.

Microwave Planar Circuits

Microwave planar circuits may also be employed especially for use innon-destructive field tests of asphalt. Planar circuits are desirablebecause they include leaky fields configurable as dielectric sensors.FIG. 15 is a plot of slotline impedance vs. asphalt density in themicrowave region, which illustrates that the complex propagationconstant of microwave radiation can be detected using many differentparameters such as complex resonant frequency, complex impedance,complex S parameters etc. The waveguiding or launching device can bemicrostrip, stripline, slotline, coplanar waveguide, circularwaveguides, rectangular waveguides, or even open or leaky waveguidessuch as surface wave launching devices. A planar circuit can measurereflection coefficients or transmission coefficients. These responsesare the direct result of the complex propagation constant in the surfaceof the asphalt and/or traveling across the surface of the asphalt aslaunched from a planar circuit. In microwave propagation, the resonatoris just one means to measure the complex propagation constant α+jβ. Thecircuits mentioned above are a means to measure the complex propagationconstant and S parameters through network analysis. The transmissionline could also be loaded with a device that interacted with thematerial. The resulting impedance shift as a function of density couldbe detected at the input port through phase or resonance using networkanalysis.

Asphalt, in particular, is a complex molecule consisting of heteroatomssuch as sulfur, nitrogen, and oxygen attached to the principle atoms ofcarbon and hydrogen. The resulting configurations are polar because ofan imbalance of electrochemical forces in the bond. The asphaltenes aregenerally dark brown, friable solids and are the most complex componentsof the cement with the highest polarity. The remaining components areresins and oils.

Because the asphalt possesses a permanent dipole moment, and is acomplex chemical, it will have a distribution of relaxation timesleading to the Cole-Davidson equationε_(r)=ε_(r∞)+(ε_(rs)−ε_(r∞))/(1+jωτ)^(α) −jσ/ωε _(o)

Where the “s” signifies the low frequency permittivity, and the ∞signifies a high frequency permittivity. The last term is included torepresent the losses from the carbon atoms or DC conductance. Thefrequency dependent term in the above equation is in the denominator,which also varies with the relaxation time τ, the inverse of therelaxation frequency. So the mobility of the molecules or ability of themolecules to rotate is defined by the relaxation time, which alsodefines the “boundary” of the materials low and high frequency response.Since asphalt is a solid or semi-solid at room temperature, and is usedas cement, it is understandable that the relaxation frequency will below, and the permittivity will not be large as a result. In the mix, theasphalt is adsorbed onto the aggregate, which further “locks” themolecular movement. Typically the dielectric constants will be less than7. When the asphalt is more liquid, the dipoles will rotate easier,increasing the relative permittivity. This is due the thermal breakingof the bonds. A measure of the strength of molecular binding is theenergy released per mole when the bonds are formed. This is theactivation energy Q. In general the dependence of the relaxationfrequency on the activation energy and temperature is f ∝ exp(−αQ/T)where T is degrees Kelvin and α is a proportional constant. As Tincreases, the relaxation frequency follows.

Because the asphalt is a heterogeneous material which can be moist,there exist free ions and charges that can form on the particularaggregates. This leads to a Maxwell-Wagner effect whereby at the lowerfrequencies, extremely large permittivities are apparent. Since this isalso a polar mechanism, this large “artificial” permittivity is alsoextremely frequency and temperature dependent. However, because of thereduced mobility or large relaxation time τ associated with thisphenomenon, most of these effects are nonexistent in the VHF band andabove. For moist soils, the dispersion can be extreme. A measure of thechange in complex permittivity at microwave frequencies and below canlead to a measure of soil porosity.

Microstrip Surface Gauge

One microwave density device is the microstrip resonator that may bemore practical in non-destructive field tests of asphalt. In this casethe microstrip is an embodiment of the DUT from FIGS. 1 and 2. One suchmicrostrip 60 is shown in FIG. 16( a). The microstrip line 60 isemployed as a waveguiding device, and since it is an open structure, thefields extend into the space above the line. This is a desirablecharacteristic for a sensor. By increasing the thickness h, of thesubstrate 62, and decreasing the permittivity of the substrate 62, moreof the field will propagate in the space above the strip as shown inFIG. 16( a). By coupling into the strip with an impedance discontinuity,the transmission line will resonate. For example, placing a material 64(FIG. 16( b)) of higher dielectric constant in the space adjacent theline, even more of the energy will exist in the material 64. This isreferred to as inverted microstrip, and as the dielectric constant ofthe material 64 is increased, the phase velocity will decrease, and theresonator will decrease its resonant frequency.

Much like the method employed above in conjunction with the cavityresonator, the Q and resonant frequency can be measured by the networkanalyzer and the effective permittivity ε′−jε″ can be found. Note thatin FIG. 16( b), the asphalt would act as the upper dielectric layer. Inone useful embodiment the microstrip is a meander line microstrip 66, asillustrated in view of FIG. 16. Ring resonators are very convenientgeometries as well. In these examples, the transmission lines wereoperated as two port structures. However one port devices can also beused such as a resonant “cavity backed dipole.” The resonance may beexclusively the dipole, or a slot antenna excited by microstrip. It mayalso be a cavity backed slot antenna or the result of a tuned shortdipole/transmission line circuit.

The microstrip resonator operates in the same principle as the waveguideresonator with a few significant differences. Like the cavity, the shiftin frequency can be measured and the Q_(L) value is

$\begin{matrix}{Q_{L} = \frac{f}{\Delta\; f}} & (20)\end{matrix}$Where Δf is usually defined by the −3 dB frequency values off maximum ofS₂₁. Using an effective real permittivity for the measurement volume ofthe resonator (a portion of the field fills the resonator), theimaginary part of the permittivity is

$\begin{matrix}{ɛ_{r\; e}^{''} = \frac{ɛ_{r\; e}^{\prime}}{Q_{Asphalt}}} & (21)\end{matrix}$As resonance occurs approximately when

$\begin{matrix}{{L = {n\frac{\lambda_{g}}{2}}}{and}} & (22) \\{\frac{c}{\sqrt{ɛ_{r\; e}}} = {f\;\lambda_{g}}} & (23)\end{matrix}$Then at the second harmonic, n=2,

$\begin{matrix}{ɛ_{r\; e}^{\prime} = \left( \frac{c}{fL} \right)^{2}} & (24)\end{matrix}$It is also a reasonable assumption that

$\begin{matrix}{\frac{1}{Q_{Asphalt}} = {\frac{1}{Q_{L2}} - \frac{1}{Q_{air}}}} & (25)\end{matrix}$Then the imaginary permittivity of the asphalt is approximated as

$\begin{matrix}{ɛ_{r\;}^{''} = \frac{ɛ_{r\; e}}{Q_{asphalt}}} & (26)\end{matrix}$

The resonant condition is perturbed from Equation (22) by the couplingstructure. For instance, the coupled transmission line would have aminimum reflection coefficient and/or a maximum transmission coefficientwhen the impedance of the load is transferred to the complex conjugateat the input reference plane. Again, this is a function of the distancebetween the input and output launching structures and the propagationconstant of the traversing wave. The launching structures could be ashort section of microstrip line fed with an SMA connector and loop orgap coupled, or an antenna capable of launching microwave fields with aparticular field orientation such as parallel to the surface of thewaveguide/asphalt interface. Parallel fields are less susceptible tosurface roughness (air gaps between the sensor and the material) thanperpendicular Electric fields.

The following sample measurements included collecting the resonantfrequency of the second harmonic, and the Q value of a microstripresonant structure similar to FIG. 17. These measurements were in therange of 680 Mhz. The second harmonic will have a maximum electric fieldat both the ends of the resonator, as well as the center. These resultswere compared to the known density, which was resolved from a nucleardensity gauge.

FIG. 18 shows the response of the resonant frequency to changes indensity. Here, back calculation of dielectric constant is not required,and it is apparent frequency and density are therefore sufficient tocalibrate the measurements. In FIGS. 19( a)–(b), both the real andimaginary frequencies were incorporated into the analysis yielding aloaded Q factor to changes in density, a two parameter curve fit. Thecurves of FIGS. 19( a)–(b) provided more consistent results. These twofigures also illustrate that both the Q value and the resonant frequencybehave in a linearly decreasing manner as the density increases for themicrostrip device.

Surface Roughness Correction

While it is preferable that the surface of the asphalt sample be smoothor have a known roughness such samples are not practicable in the fieldor in the laboratory. Accordingly, it is desirable to include aroughness correction in conjunction with either or both of theembodiments of a density gauge described above. One embodiment of aroughness gauge is an electromagnetic system having two permittivitysensors. One sensor reads shallow (˜0.25″) depths below the surface, anda second sensor reads deeper (>2″), but also includes the shallow layerin its measurement volume. The sensor is electromagnetic, and uses thepermittivity of the material to calculate the roughness. In oneembodiment, the shallow sensor ran at 50 Mhz and was a fringing fieldcapacitance sensor such as disclosed in U.S. Pat. No. 6,803,771,incorporated herein by reference, while the deeper sensor was amicrowave traveling wave type resonator, such as illustrated in FIG. 17.If the mix is constant over the two-inch depth, then it is reasonable toexpect that the surface texture will affect the shallow sensor more thanthe deep sensor.

Since it is extremely difficult to obtain perfect asphalt cores orslabs, a finite element simulation model validates the measurementsdifferences. One embodiment is shown in FIG. 20 wherein the microwavedevice is a covered microstrip line 70 on a substrate 72 of thickness hplace on asphalt 76 having a roughness 74. By solving for thepropagation constant of the strip, the effective permittivity of theentire structure is determined. Resonant frequencies can then be foundusing the actual device dimensions.

This microstrip resonator 70 is 0.304 m long, 9.92 mm wide with gapcoupling. A 1/32″ thick epoxy FR-4 cover is glued directly on the stripas shown in FIG. 20. It was determined using the HP dielectric probethat this cover had ε=4.0. The substrate 72 was 0.125″ thick and isPolyflon Norclad, ε=2.55. In the simulations, the rough layer 74 washeld at 0.25 inches deep, but the dielectric constant of this layer wasadjusted from equal the underlying asphalt 76, to 50% less the asphaltvalue. Note that the rough layer has a permittivity that is equal to orless than the asphalt, because of the surface voids represented with thejagged line at the asphalt interface in FIG. 19.

Five sets of simulations were run at 600 Mhz with different asphaltpermittivities. These were ε=3.0, 4.4, 6.17, and 7.11. These are allvalues of materials that are available in the lab. For each set ofsimulations, the asphalt dielectric constant was held constant, whilethe rough layer permittivity was changed. For each simulation, the roughlayer was adjusted to correspond to a percent of the asphalt 6, mainly100%, 90%, 80%, 60%, and 50%, except for the asphalt ε=6.17, where 68%was used. The 68% was chosen so that the rough layer simulation wouldhave a permittivity corresponding to FR-4 for laboratory verification.

The effective dielectric constant, ε_(re), of the (combined) substrate,cover, roughness layer, and asphalt was determined. The simulations werecalibrated to remove the effects of the substrate by simulating theresonator on the known materials and obtaining the resonant frequencyand Q value. A calibration curve and equation of the resonant frequencyvs. the external material properties can be determined. Hence, absolutepermittivities of any flat sample can now be obtained. From the model,the following results were obtained.

TABLE 5 ASPHALT ε = 7.11 Roughness layer 1/4 in. thick. Asphalt = 7.11εre of structure Composite Permittivity Rough = 7.11 3.43 7.112 Rough =6.4 3.35 6.661 Rough = 5.7 3.27 6.209 Rough = 4.27 3.08 5.137 Rough =3.55 2.97 4.517 Rough = 7.725*ε_(re) − 19.46 ASPHALT ε = 6.17 Asphalt =6.17 εre Composite Permittivity Rough = 6.17 3.26 6.153 Rough = 5.553.19 5.758 Rough = 4.94 3.12 5.363 Rough = 4.2 3.03 4.855 Rough = 3.12.87 3.952 Rough = 7.875*ε_(re) − 19.57 ASPHALT ε = 4.4 Asphalt = 4.4εre Composite Permittivity Rough = 4.4 2.95 4.404 Rough = 3.96 2.904.122 Rough = 3.52 2.84 3.783 Rough = 2.64 2.72 3.106 Rough = 2.2 2.662.768 Rough = 7.50*ε_(re) − 17.76 ASPHALT ε = 3.0 Asphalt = 3.0 εreComposite Permittivity Rough = 3.0 2.7 2.993 Rough = 2.7 2.66 2.768Rough = 2.4 2.62 2.542 Rough = 1.8 2.54 2.091 Rough = 1.5 2.49 1.808Rough = 7.22*ε_(re) − 16.49

Note that the first assignment to the rough layer is the value of theasphalt (2^(nd) row of each table). With both layers assigned the samepermittivity, the simulated calibration of a homogeneous smooth sampleis accomplished by plotting the ε_(re) vs. the assigned externalmaterial property. Combining with the data of Table 1, we obtain thecomposite permittivity,ε_(external)=5.642*ε_(re)−12.24.  (27)Where “external” represents the permittivity of the material excludingthe microstrip substrate and cover. This equation is shown in FIG. 21and represents the data of row 2 in the table for each ε_(r)=7.11, 6.17,4.4, 3.0. Note that FIG. 21 relates only to the perfectly smooth andhomogenous surface. In other words, the asphalt layer and the roughlayer have been assigned the same value. As the surface becomestextured, the permittivity of the rough layer will decrease. This willdecrease the effective permittivity of the entire structure, even if theasphalt layer does not change. Equation (27) is still used to find thiscomposite ε_(r). For the two-sensor system, the shallow sensor will havea larger reduction in permittivity than the deep sensor. It is thisprinciple upon which the correction is based. The surface roughnesscorrection should allow for the shallow layer permittivity to decrease,yet still obtain a good estimate of the permittivity of the deeperlayer.

It is noteworthy that different permittivity combinations of asphalt androughness can lead to the same effective or composite dielectricconstant. If the asphalt ε_(r)=7.11 and the surface is ε_(r)=4.27 thentable 1 shows ε_(external)=5.137 as measured by the deep sensor.However, if the asphalt ε_(r)=6.14 and it is relatively smooth pavementhas a surface value of ε_(r)=4.68, then the ε_(external) is also 5.137;according to table 1. A lower valued asphalt with a smoother surfacecould result in a higher effective permittivity than the higher valuedasphalt with the rougher surface. Therefore, in one method a ratio ofthe shallow sensor/deep sensor result may used for correction. This way,for smooth surfaces, the ratio will be 1 and no correction is obtained.However, for a rougher surface, the ratio will be less than 1, butgreater than zero, and a larger correction is obtained. For instance, inthe example above, the ratios are 4.27/5.137=0.83, and 4.68/5.137=0.91.The second example with a factor of 0.91 is smoother than the 0.83situation, and the correction will be much less.

In the finite element simulations, the results ε_(re) were converted tothe external or “Top” values using Equation (27) of FIG. 21. To obtain a“first guess” of the deep layer dielectric constant. One possiblecorrection for the actual permittivity could beε₂=(ε_(x)−ε₁)*e ^((K*h1/w))+ε₁  (28)WhereK=(w/h ₁)*ln(2/(1+1/x))x=√(1+12h/w),  (29)w is the strip width ⅜″, h₁=¼″ is the thickness of the rough layer, andε_(x), ε₂ are the independent permittivities of the external layer,rough layer and deep layer respectively. Notice that the external layerε_(x) is the composite response of both the rough and deep layer, as“measured” by the deep sensor. In Equation (28), as h₁ becomes small,ε₂=ε_(x).

Relationships between the external dielectric constant ε_(x) and thedeep layer permittivity were also considered. A guess of ε₂ is foundusing Equation (28). Taking a known permittivity value for the asphaltdeep layer, and the deep layer “first guess,” Equation (28),experimental results for the deep (D) and shallow (S) sensors can becompared to the ratio of the first guess (ε₂) and known asphalt values(ε_(known)). The finite element analysis calculated compositepermittivities were used to obtain the values for the deep and shallowsensors. The experimental results indicate that the ratio of themeasured shallow and deep sensors is related to the ratio of the deeplayer and the composite layer. The experimental relationship is linearas shown in FIG. 22, where in this example the slope is 1.5187 and theintercept is 0.5184.

For example, suppose that the shallow sensor measured 5.55 and the deepsensor obtained the composite external dielectric constant of 5.75.Using slope and intercept from “first guess” of Equation 28, then thevalue of the asphalt layer would be (see the 6.17 asphalt layer in Table1)(5.55/5.75)*1.5187−0.5184=0.9454Asphalt=5.86/0.9454=6.20.Alternatively, it may not be necessary to use the “first guess,” andonly consider the measured values of deep and shallow sensors ascompared to the known value. Therefore, the linear relationship betweenthe ratio of the deep sensor ε_(x), and known deep permittivity can beplotted versus shallow sensor and deep sensor ratio values, resulting ina slope of 1.828 and an intercept of 0.8316.

Another roughness correction approach is to use actual core measurementsto calculate the ratios of the known bulk permittivity (ε_(known)) vs.the deep and shallow sensor ratios. For example, since there are twounknowns in the linear equation of FIG. 22, measurements on a smoothside and a rough side of the same core could be used for the roughnesssolution. One experimental core a very smooth cleaved side, the “X”side, and a normal rough side, the “O” side. In this example, the bulkpermittivity (ε_(known)) was measured in the microwave cavity and wasfound to be 5.75. This bulk number was assumed to be the permittivityjust under the surface. Using the (D) deep microwave sensor and the (S)shallow UHF sensor the following dielectric constants were obtained, inTable 5.

TABLE 5 Sensor “X” = Smooth “O” = Rough Deep 5.7 5.15 Shallow 5.6 4.24S/D 5.6/5.7 4.24/5.15 D/(ε_(known)) 5.7/5.75 5.15/5.75From this sample, two equations can be determined such that this sampleyields lines“X”0.9913=0.9820*m+b“O”0.8956=0.8233*m+b.Solving these simultaneously leads toD/Z=S/D*0.6027+0.4  (30)Therefore, by measuring the material with a deep and a shallow sensor,the composite permittivity found with the deep sensor, D, can beseparated into the two layers, S and the unknown Z, as shown in Equation(30).

Another approach is to measure the two ends of an un-cleaved core, whichtypically have different textures, and to use a least squares method tocalculate the average slope and intercept of Equation (30). Instead ofusing one core with one artificial smooth face and a normal face, a morerealistic approach may be to measure many field mixes made into cores.In this method, it is assumed that the core is homogeneous, and that thesame permittivity should be attained on each face of the sample. Firstthe two sides are labeled as “X” and “O” sides. Then each sample isplaced in a resonant closed cavity and the frequency and bandwidthshifts are recorded. A finite element analysis is run on the cavity andcore dimensions, and the real part of the permittivity is found. Withthis method, the bulk permittivity (ε_(known)) is the final result.Next, both the deep sensor (D) and the shallow sensor (S) are used tomeasure the permittivity of the “X” side and the “O” side of the puck.It is assumed that any differences in these two sides are due to thedifferent texture or surface properties of the puck on each face. Once agood statistical population of cores is measured, a table of S/D,D/(ε_(known)) is formed for each sample on each side “X” and “O”.

For example, each face is measured with the shallow sensor (S-200 @ 50MHz) and the deep sensor (BS @ 600 MHz). The ratios S/D for both faces(X, O) are then plotted against the ratios D/C for each core in hopesthat a correlation exists between these results, illustrated in FIG. 23.

The regression analysis indicates thatD/(ε_(known))=0.405*S/D+0.566  (31)With an r² of 0.482. The correlation coefficient may be improved withsensors that have less orientational properties and a better defineddepth dependency, as the regression may indicate that the shallow sensorreads deeper than expected.

Accordingly, the two-layers can be simulated using finite elements toobtain equations that describe the calculation of the deep layerdielectric from the shallow and deep measurements. These equations wouldthen be implemented and the coefficients adjusted for a calibration.Likewise, calibration may be empirical by placing the plates on knowndielectric materials.

In another modification of the surface roughness correction meter, thecavity measurements can be used to obtain the loss properties of theasphalt field or laboratory sample. The loss can be converted toconductivity, and a skin depth or depth of penetration can becalculated. It is well known that this depth is a function of frequency.Hence, by proper selection of frequency, an automatically depthdependent measurement can be obtained. In a slightly differentembodiment, for surface waves, as the frequency increases, the surfacewaves excited by the sensors will penetrate less. This is due to boththe losses due to the conductance, and the boundary solution toMaxwell's Equations for TE or TM type surface waves as known by thosefamiliar in the art.

Many modifications and other embodiments of the inventions set forthherein will come to mind to one skilled in the art to which theseinventions pertain having the benefit of the teachings presented in theforegoing descriptions and the associated drawings. Therefore, it is tobe understood that the inventions are not to be limited to the specificembodiments disclosed and that modifications and other embodiments areintended to be included within the scope of the appended claims.Although specific terms are employed herein, they are used in a genericand descriptive sense only and not for purposes of limitation.

1. A method of determining a property of a sample from a responsethereof to a field, the method comprising: exciting a resonant deviceacross a contiguous radio frequency band, the radio frequency bandextending up to and including a microwave frequency range, so as toproduce an electromagnetic field having at least a first microwavefrequency resonance mode, the resonant device being configured tointeract with a pavement material sample comprising one of a cohesivesoil mixture and an asphalt mixture, wherein the exciting step isperformed according to a skin depth property of the pavement materialsample such that the resonant device interacts with the pavementmaterial sample at a selected depth therein, the resonance device beingexcited at lower frequencies within the radio frequency band for agreater interaction depth and at higher frequencies within the radiofrequency band for a lesser interaction depth with the pavement materialsample; comparing a first response of the resonant device to theinteraction with the pavement material sample across the radio frequencyband, to a second response of the resonant device to an interaction witha control sample across the radio frequency band, so as to determine afrequency response of the pavement material sample to theelectromagnetic field across the radio frequency band; and correlatingthe frequency response of the pavement material sample across the radiofrequency band to a material property of the pavement material sample,the material property comprising one of a moisture content and amoisture-less density of the pavement material sample.
 2. The methodaccording to claim 1, further comprising: exciting the resonant deviceacross the contiguous radio frequency band, so as to produce theelectromagnetic field having at least the first microwave frequencyresonance mode within the radio frequency band, the resonant devicebeing configured to interact with the control sample comprising acalibration material having a known physical property selected from thegroup consisting of density, volume, specific gravity, and porosity;determining the second response of the resonant device to theinteraction with the control sample across the radio frequency band soas to determine a frequency response of the control sample to theelectromagnetic field across the radio frequency band; and correlatingthe frequency response of the control sample to the known physicalproperty thereof, so as to provide a calibration curve with respect tothe control sample across the radio frequency band, for comparison tothe first response of the resonant device to the interaction with thepavement material sample across the radio frequency band.
 3. The methodaccording to claim 1, wherein exciting a resonant device furthercomprises exciting a resonant device comprising one of a planarmicrowave circuit, a microstrip, a slotline, a dipole, a monopole, andan inverted microstrip so as to produce the electromagnetic field. 4.The method according to claim 1, wherein comparing the first response tothe second response further comprises comparing the first response tothe second response, each of the first and second responses beingcorrespondingly selected from the group consisting of a scatteringparameter, an admittance matrix parameter, an impedance matrixparameter, a quality factor, complex permittivity, a complex propagationconstant, and a voltage standing wave ratio.
 5. The method according toclaim 1, wherein comparing a first response to a second response todetermine a frequency response of the pavement material sample furthercomprises comparing the first response to the second response todetermine a permittivity characteristic as the frequency response of thepavement material sample to the electromagnetic field across the radiofrequency band.
 6. The method according to claim 1, wherein exciting aresonant device farther comprises exciting the resonant device so as toproduce the electromagnetic field having at least the first microwavefrequency resonance mode such that, for at least the first microwavefrequency resonance mode, a vector associated with the electromagneticfield is substantially parallel to a rough surface of the pavementmaterial sample so as to minimize a surface roughness error in thedetermined frequency response of the pavement material sample.
 7. Amethod of determining a density of a sample engaged with a waveguide,the method comprising: exciting a resonant device across a contiguousradio frequency band, the radio frequency band extending up to andincluding a microwave frequency range, so as to produce anelectromagnetic field, having at least a first microwave frequencyresonance mode, configured to interact with a microwave waveguide and apavement material engaged with the microwave waveguide, wherein theexciting step is performed according to a skin depth property of thepavement material sample such that the resonant device interacts withthe pavement material sample at a selected depth therein, the resonancedevice being excited at lower frequencies within the radio frequencyband for a greater interaction depth and at higher frequencies withinthe radio frequency band for a lesser interaction depth with thepavement material sample; comparing a first response of the resonantdevice to the interaction with the microwave waveguide, to a secondresponse of the resonant device to an interaction with the pavementmaterial sample engaged with the microwave waveguide, across the radiofrequency band; determining a difference between the first and secondresponses of the resonant device to provide a frequency response of thepavement material sample to the electromagnetic field; and correlatingthe frequency response of the pavement material sample to theelectromagnetic field across the radio frequency band to one of amoisture content and a moisture-less density of the pavement materialsample.
 8. The method according to claim 7, wherein comparing a firstresponse to a second response of the resonant device further comprisescomparing the first response to the second response, each of the firstand second responses being correspondingly selected from the groupconsisting of a scattering parameter, an admittance matrix parameter, animpedance matrix parameter, a quality factor, complex permittivity, acomplex propagation constant, and a voltage standing wave ratio.
 9. Themethod according to claim 7, wherein determining a difference betweenthe first and second responses to provide a frequency response of thepavement material sample further comprises determining a differencebetween the first and second responses so as to determine a permittivitycharacteristic as the frequency response of the pavement material sampleto the electromagnetic field across the radio frequency band.
 10. Themethod according to claim 7, wherein exciting a resonant device furthercomprises exciting a resonant device comprising one of a planarmicrowave circuit, a microstrip, a slotline, a dipole, a monopole, andan inverted microstrip so as to produce the electromagnetic field. 11.The method according to claim 7, wherein exciting a resonant devicefurther comprises exciting the resonant device so as to produce theelectromagnetic field having at least the first microwave frequencyresonance mode such that, for at least the first microwave frequencyresonance mode, a vector associated with the electromagnetic field issubstantially parallel to a rough surface of the pavement materialsample so as to minimize a surface roughness error in the determinedfrequency response of the pavement material sample.
 12. The methodaccording to claim 7, farther comprising correcting one of the moisturecontent and the moisture-less density of the pavement material samplefor an effect of temperature of the pavement material sample.
 13. Amethod of obtaining a density of a pavement material with a planarmicrowave circuit, the method comprising: obtaining a calibration dataset comprising a plurality of calibration frequency responses of theplanar microwave circuit as a function of density of a calibrationmaterial; placing the planar microwave circuit over a layer of thepavement material; exciting the planar microwave circuit across acontiguous frequency band including a microwave frequency range, so asto produce an electromagnetic field of a first mode with the planarmicrowave circuit about the pavement material, wherein the first mode isselected such that the electromagnetic field's electric fields aresubstantially parallel to a rough surface to minimize surface roughnesseffects, and wherein the exciting step is performed according to a skindepth property of the pavement material sample such that theelectromagnetic field generated by the planar microwave circuitinteracts with the pavement material sample at a selected depth therein,the planar microwave circuit being excited at lower frequencies withinthe frequency band for a greater interaction depth and at higherfrequencies within the frequency band for a lesser interaction depthwith the pavement material sample; measuring a sample frequency responseof the pavement material; and correlating the sample frequency responseto a density based on the calibration data set.
 14. The method accordingto claim 13, the step of correlating the sample frequency responsefarther comprising correlating the sample response selected from thegroup consisting of scattering parameters, admittance matrix parameters,impedance matrix parameters, quality factors, complex permittivity,complex propagation constant, and voltage standing wave ratio.
 15. Themethod according to claim 13, further comprising calculating apermittivity from the sample frequency response to correlate the samplefrequency response to the density.
 16. The method according to claim 13,wherein the step of generating comprises generating a microwaveelectromagnetic field of a first mode about a pavement material selectedfrom the group consisting of soil, sand, aggregate, asphalt, and cement.17. The method according to claim 13, further comprising: obtaining asecond calibration data set comprising a plurality of calibration sampleparameters from the planar circuit wherein the calibration sampleparameters are selected from the group consisting of scatteringparameters, voltage standing wave ratio, quality factors, complexpermittivity, and complex propagation constants; measuring a sampleparameter of the pavement material; and correlating the sample parameterto a density of the pavement material sample based on the secondcalibration data set.
 18. The method according to claim 13, furthercomprising correcting the density based upon the effects of temperature.